The previous discussion solved for the motion of an ideal mass
striking an ideal string of infinite length. We now investigate the
same model from the string's point of view. As before, we will be
interested in a digital waveguide (sampled traveling-wave) model of
the string, for efficiency's sake (Chapter 6), and we
therefore will need to know what the mass ``looks like'' at the end of
each string segment. For this we will find that the impedance
description (§7.1) is especially convenient.

Figure 9.15:Physical model of
mass-string collision after time 0. The mass is drawn as having a
finite diameter for conceptual clarity. However, the model is
formulated for the limit as the diameter approaches zero in the
figure (bringing all three forces together to act on a single
mass-string junction point). In other words, we assume a point
mass.

Let's number the string segments to the left and right of the mass by
1 and 2, respectively, as shown in Fig.9.15. Then
Eq.(9.8) above may be written

(10.11)

where denotes the force applied by string-segment 1 to the
mass (defined as positive in the ``up'', or positive- direction),
is the force applied by string-segment 2 to the mass
(again positive upwards), and denotes the inertial force applied by
the mass to both string endpoints (where again, a positive force
points up).

Figure 9.16:
Depiction of a string
segment with negative slope (center), pulling up to the right and down
to the left. (Horizontal force components are neglected.)

As shown in Fig.9.16, a negative string slope pulls ``up''
to the right. Therefore, at the mass point we have
, where denotes the position of the
mass along the string. On the other hand, the figure also shows that
a negative string slope pulls ``down'' to the left, so that
. In summary, relating the forces we
have defined for the mass-string junction to the force-wave variables
in the string, we have

The inertial force of the mass is
because the mass
must be accelerated downward in order to produce an upward reaction
force. The signs of the two string forces follow from the definition
of force-wave variables on the string, as discussed above.

The force relations can be checked individually. For string 1,

states that a positive slope in the string-segment to the left of the
mass corresponds to a negative acceleration of the mass by the endpoint
of that string segment. Similarly, for string 2,

says that a positive slope on the right accelerates the mass upwards.
Similarly, a negative slope pulls ``up'' to the right and ``down'' to
the left, as shown in Fig.9.16 above.

Now that we have expressed the string forces in terms of string
force-wave variables, we can derive digital waveguide models by
performing the traveling-wave decompositions
and
and using the Ohm's law relations
and
for (introduced above near
Eq.(6.6)).